General relativity, integration over a manifold exercise

The problem I am facing is 2.9 in Sean Carroll's book on general relativity (Geometry and Spacetime) I should note that I am not studying this formally and so a full solution would not be unwelcome, though I understand that forum policy understandably prohibits it. The full problem is stated as

a.) Evaluate [tex]d*F=*J[/tex]
b.) What is the two-form [tex]F[/tex] equal to?
c.) What are the electric and magnetic fields equal to for this solution?
d.) Evaluate [tex]\int_V d*F[/tex], where [tex]V[/tex] is a ball of radius [tex]R[/tex] in Euclidean three-space at a fixed moment of time.

2. Relevant equations

In the above the asterisk denotes the Hodge dual and the [tex]d[/tex] denotes the exterior derivative. The definitions of these operators should be well known.

3. The attempt at a solution

I think I have the first three parts solved. For each part a.) I arrived at the lengthy result of

Verification of these would be appreciated but I am especially confused about the last part d.). In particular the integrand is a three-form and I have no idea how to integrate this and how to transform the integral into a familiar volume integral that can be computed by standard multivariable calculus.